High order Fuchsian equations for the square lattice Ising model: $\chi^{(6)}$

TL;DR

This paper derives and factorizes a high-order Fuchsian differential equation for the six-particle contribution to the magnetic susceptibility of the square lattice Ising model, revealing its structure and relation to elliptic integrals.

Contribution

It introduces a method to generate and factorize the differential equation for , showing its structure and connection to elliptic integrals, extending previous work on  and  contributions.

Findings

The differential operator for  splits into smaller factors.

The left-most factor is equivalent to the symmetric fifth power of an elliptic integral operator.

The factors in the direct sum are reconstructed exactly from series data.

Abstract

This paper deals with $\tilde{\chi}^{(6)}$, the six-particle contribution to the magnetic susceptibility of the square lattice Ising model. We have generated, modulo a prime, series coefficients for $\tilde{\chi}^{(6)}$. The length of the series is sufficient to produce the corresponding Fuchsian linear differential equation (modulo a prime). We obtain the Fuchsian linear differential equation that annihilates the "depleted" series $\Phi^{(6)}=\tilde{\chi}^{(6)} - {2 \over 3} \tilde{\chi}^{(4)} + {2 \over 45} \tilde{\chi}^{(2)}$. The factorization of the corresponding differential operator is performed using a method of factorization modulo a prime introduced in a previous paper. The "depleted" differential operator is shown to have a structure similar to the corresponding operator for $\tilde{\chi}^{(5)}$. It splits into factors of smaller orders, with the left-most factor of order six being equivalent to the symmetric fifth power of the linear differential operator corresponding to the elliptic integral $E$. The right-most factor has a direct sum structure, and using series calculated modulo several primes, all the factors in the direct sum have been reconstructed in exact arithmetics.